Dynamics of the Picking transformation on integer partitions

Abstract : This paper studies a conservative transformation defined on families of finite sets. It consists in removing one element from each set and adding a new set composed of the removed elements. This transformation is conservative in the sense that the union of all sets of the family always remains the same. We study the dynamical process obtained when iterating this deterministic transformation on a family of sets and we focus on the evolution of the cardinalities of the sets of the family. This point of view allows to consider the transformation as an application defined on the set of all partitions of a fixed integer (which is the total number of elements in the sets). We show that iterating this particular transformation always leads to a heterogeneous distribution of the cardinalities, where almost all integers within an interval are represented. We also tackle some issues concerning the structure of the transition graph which sums up the whole dynamics of this process for all partitions of a fixed integer.
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Michel Morvan and Éric Rémila. Discrete Models for Complex Systems, DMCS'03, 2003, Lyon, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AB, Discrete Models for Complex Systems (DMCS'03), pp.43-56, 2003, DMTCS Proceedings
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Thi Ha Duong Phan, Eric Thierry. Dynamics of the Picking transformation on integer partitions. Michel Morvan and Éric Rémila. Discrete Models for Complex Systems, DMCS'03, 2003, Lyon, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AB, Discrete Models for Complex Systems (DMCS'03), pp.43-56, 2003, DMTCS Proceedings. 〈hal-01183319〉

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